probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

, an outcome is a possible result of an

experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs whe ...

or trial. Each possible outcome of a particular experiment is unique, and different outcomes are

mutually exclusive In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...

(only one outcome will occur on each trial of the experiment). All of the possible outcomes of an experiment form the elements of a

sample space In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...

. For the experiment where we flip a coin twice, the four possible ''outcomes'' that make up our ''sample space'' are (H, T), (T, H), (T, T) and (H, H), where "H" represents a "heads", and "T" represents a "tails". Outcomes should not be confused with ''

events Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of ev ...

'', which are (or informally, "groups") of outcomes. For comparison, we could define an event to occur when "at least one 'heads'" is flipped in the experiment - that is, when the outcome contains at least one 'heads'. This event would contain all outcomes in the sample space except the element (T, T).

Sets of outcomes: events

Since individual outcomes may be of little practical interest, or because there may be prohibitively (even infinitely) many of them, outcomes are grouped into sets of outcomes that satisfy some condition, which are called "

." The collection of all such events is a sigma-algebra. An event containing exactly one outcome is called an

elementary event In probability theory, an elementary event, also called an atomic event or sample point, is an event which contains only a single outcome in the sample space. Using set theory terminology, an elementary event is a singleton. Elementary events a ...

. The event that contains all possible outcomes of an experiment is its

. A single outcome can be a part of many different events. Typically, when the sample space is finite, any subset of the sample space is an event (that is, all elements of the

power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...

of the sample space are defined as events). However, this approach does not work well in cases where the sample space is

uncountably infinite In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...

(most notably when the outcome must be some

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

). So, when defining a

probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...

it is possible, and often necessary, to exclude certain subsets of the sample space from being events.

Probability of an outcome

Outcomes may occur with probabilities that are between zero and one (inclusively). In a discrete probability distribution whose

is finite, each outcome is assigned a particular probability. In contrast, in a

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

distribution, individual outcomes all have zero probability, and non-zero probabilities can only be assigned to ranges of outcomes. Some "mixed" distributions contain both stretches of continuous outcomes and some discrete outcomes; the discrete outcomes in such distributions can be called atoms and can have non-zero probabilities. Under the measure-theoretic definition of a

, the probability of an outcome need not even be defined. In particular, the set of events on which probability is defined may be some

σ-algebra In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...

S

and not necessarily the full

Equally likely outcomes

In some

s, it is reasonable to estimate or assume that all outcomes in the space are equally likely (that they occur with equal

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...

). For example, when tossing an ordinary coin, one typically assumes that the outcomes "head" and "tail" are equally likely to occur. An implicit assumption that all outcomes are equally likely underpins most

randomization Randomization is the process of making something random. Randomization is not haphazard; instead, a random process is a sequence of random variables describing a process whose outcomes do not follow a deterministic pattern, but follow an evolution d ...

tools used in common games of chance (e.g. rolling

dice Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing ...

, shuffling cards, spinning tops or wheels, drawing lots, etc.). Of course, players in such games can try to cheat by subtly introducing systematic deviations from equal likelihood (for example, with

marked cards Card marking is the process of altering playing cards in a method only apparent to marker or conspirator, such as by bending or adding visible marks to a card. This allows different methods for card sharps to cheat or for magicians to perform ma ...

, loaded or shaved dice, and other methods). Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely. However, there are experiments that are not easily described by a set of equally likely outcomes— for example, if one were to toss a thumb tack many times and observe whether it landed with its point upward or downward, there is no symmetry to suggest that the two outcomes should be equally likely.

References

External links

*{{Commonscat-inline Experiment (probability theory)

Sets of outcomes: events

Probability of an outcome

Equally likely outcomes

See also

References

External links